hyperperfect number

hyper- +‎ perfect number.

hyperperfect number (plural hyperperfect numbers)

1. (mathematics, number theory) Any natural number n for which, for some positive integer k, n = 1 + k(σ(n) - n - 1), where σ(n) is the sum of the positive divisors of n.
   • 1966, American Mathematical Society Translations, page 258,

     […] the asymptotic density of all hyperperfect numbers, that is, numbers m for which m | σ(m), is equal to zero.

   • 1974, William Judson LeVeque, editor, Reviews in number theory, as printed in Mathematical reviews, 1940 through 1972, volumes 1-44 inclusive‎, volume 1, page 107:

     The rank of a hyperperfect number N is the ratio of divisor sum to N (which equals 2 for perfect numbers).

   • 1999, James J. Tattersall, Elementary Number Theory in Nine Chapters‎, page 144:

     In 1974, Daniel Minoli and Robert Bear described a number of properties of hyperperfect numbers.

Note that hyperperfect numbers are more numerous than perfect numbers (since all perfect numbers are hyperperfect).

Making the relationship with perfect number slightly clearer, the defining equation is sometimes rendered as ${\displaystyle n=1+k\sum _{i}d_{i}}$, where the terms ${\displaystyle d_{i}}$ are the proper divisors of n (in this context, excluding both 1 and n). n is also said to be a k-hyperperfect number. A 1-hyperperfect number (or unitary hyperperfect number) is a perfect number.

• -hyperperfect number

• perfect number
• superperfect number